Fabry–Pérot interferometer In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces (i.e: thin mirrors). Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899.[1][2][3] Etalon is from the French étalon, meaning "measuring gauge" or "standard".[4] Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry–Pérot interferometers. The device is called an interferometer when the distance between the two surfaces (and with it the resonance length) can be changed, and etalon when the distance is fixed (however, the two terms are often used Interference fringes, showing fine structure, from a Fabry– interchangeably). Pérot etalon. The source is a cooled deuterium lamp. Contents Basic description Applications Theory Resonator losses, outcoupled light, resonance frequencies, and spectral line shapes Generic Airy distribution: The internal resonance enhancement factor Other Airy distributions Airy distribution as a sum of mode profiles Characterizing the Fabry-Pérot resonator: Lorentzian linewidth and finesse Scanning the Fabry-Pérot resonator: Airy linewidth and finesse Frequency-dependent mirror reflectivities Fabry-Pérot resonator with intrinsic optical losses Description of the Fabry-Perot resonator in wavelength space See also Notes References External links Basic description The heart of the Fabry–Pérot interferometer is a pair of partially reflective glass optical flats spaced micrometers to centimeters apart, with the reflective surfaces facing each other. (Alternatively, a Fabry–Pérot etalon uses a single plate with two parallel reflecting surfaces.) The flats in an interferometer are often made in a wedge shape to prevent the rear surfaces from producing interference fringes; the rear surfaces often also have an anti-reflective coating. In a typical system, illumination is provided by a diffuse source set at the focal plane of a collimating lens. A focusing lens after the pair of flats would produce an inverted image of the source if the flats were not present; all light emitted from a point on the source is focused to a single point in the system's image plane. In the accompanying illustration, only one ray emitted from point A on the source is traced. As the ray passes through the paired flats, it is multiply reflected to produce multiple transmitted rays which are collected by the focusing lens and brought to point A' on the screen. The complete interference pattern takes the appearance of a set of concentric rings. The sharpness of the rings depends on the reflectivity of the flats. If the reflectivity is high, resulting in a high Q factor, monochromatic light produces a set of narrow bright rings against a dark background. A Fabry–Pérot interferometer with high Q is said to have high finesse. Applications Telecommunications networks employing wavelength division multiplexing have add-drop multiplexers with banks of miniature tuned fused silica or diamond etalons. These are small iridescent cubes about 2 mm on a side, mounted in small high- precision racks. The materials are chosen to maintain stable mirror-to-mirror distances, and to keep stable frequencies even when the temperature varies. Diamond is preferred because it has greater heat conduction and still has a low coefficient of expansion. In 2005, some telecommunications equipment companies began using solid etalons that are themselves optical fibers. This eliminates most mounting, alignment and cooling Fabry–Pérot interferometer, using a pair of partially difficulties. reflective, slightly wedged optical flats. The wedge angle is Dichroic filters are made by depositing a series of etalonic layers highly exaggerated in this illustration; only a fraction of a on an optical surface by vapor deposition. These optical filters degree is actually necessary to avoid ghost fringes. Low- usually have more exact reflective and pass bands than finesse versus high-finesse images correspond to mirror absorptive filters. When properly designed, they run cooler than absorptive filters because they can reflect unwanted reflectivities of 4% (bare glass) and 95%. wavelengths. Dichroic filters are widely used in optical equipment such as light sources, cameras, astronomical equipment, and laser systems. Optical wavemeters and some optical spectrum analyzers use Fabry–Pérot interferometers with different free spectral ranges to determine the wavelength of light with great precision. Laser resonators are often described as Fabry–Pérot resonators, although for many types of laser the reflectivity of one mirror is close to 100%, making it more similar to a Gires– Tournois interferometer. Semiconductor diode lasers sometimes use a true Fabry–Pérot geometry, due to the difficulty of coating the end facets of the chip. Quantum cascade lasers often employ Fabry-Pérot cavities to sustain lasing without the need for any facet coatings, due to the high gain of the active region.[5] Etalons are often placed inside the laser resonator when constructing single-mode lasers. Without an etalon, a laser will generally produce light over a wavelength range corresponding to a number of cavity modes, which are similar to Fabry–Pérot modes. Inserting an etalon into the laser cavity, with well-chosen finesse and free-spectral range, can suppress all cavity modes except for one, thus changing the operation of the laser from A commercial Fabry-Perot device multi-mode to single-mode. Fabry–Pérot etalons can be used to prolong the interaction length in laser absorption spectrometry, particularly cavity ring-down, techniques. A Fabry–Pérot etalon can be used to make a spectrometer capable of observing the Zeeman effect, where the spectral lines are far too close together to distinguish with a normal spectrometer. In astronomy an etalon is used to select a single atomic transition for imaging. The most common is the H-alpha line of the sun. The Ca-K line from the sun is also commonly imaged using etalons. In gravitational wave detection, a Fabry–Pérot cavity is used to store photons for almost a millisecond while they bounce up and down between the mirrors. This increases the time a gravitational wave can interact with the light, which results in a better sensitivity at low frequencies. This principle is used by detectors such as LIGO and Virgo, which consist of a Michelson interferometer with a Fabry–Pérot cavity with a length of several kilometers in both arms. Smaller cavities, usually called mode cleaners, are used for spatial filtering and frequency stabilization of the main laser. Theory Resonator losses, outcoupled light, resonance frequencies, and spectral line shapes The spectral response of a Fabry-Pérot resonator is based on interference between the light launched into it and the light circulating in the resonator. Constructive interference occurs if the two beams are in phase, leading to resonant enhancement of light inside the resonator. If the two beams are out of phase, only a small portion of the launched light is stored inside the resonator. The stored, transmitted, and reflected light is spectrally modified compared to the incident light. Assume a two-mirror Fabry-Pérot resonator of geometrical length , homogeneously filled with a medium of refractive index . Light is launched into the resonator under normal incidence. The round-trip time of light travelling in the resonator with speed , where is the speed of light in vacuum, and the free spectral range are given by The electric-field and intensity reflectivities and , respectively, at mirror are If there are no other resonator losses, the decay of light intensity per round trip is quantified by the outcoupling decay-rate constant and the photon-decay time of the resonator is then given by[6] With quantifying the single-pass phase shift that light exhibits when propagating from one mirror to the other, the round-trip phase shift at frequency accumulates to[6] Resonances occur at frequencies at which light exhibits constructive interference after one round trip. Each resonator mode with its mode index , where is an integer number in the interval [ , …, −1, 0, 1, …, ], is associated with a resonance frequency and wavenumber , Two modes with opposite values and of modal index and wavenumber, respectively, physically representing opposite propagation directions, occur at the same absolute value of frequency.[7] The decaying electric field at frequency is represented by a damped harmonic oscillation with an initial amplitude of and a decay-time constant of . In phasor notation, it can be expressed as[6] Fourier transformation of the electric field in time provides the electric field per unit frequency interval, Each mode has a normalized spectral line shape per unit frequency interval given by whose frequency integral is unity. Introducing the full-width-at-half-maximum (FWHM) linewidth of the Lorentzian spectral line shape, we obtain expressed in terms of either the half-width-at-half-maximum (HWHM) linewidth or the FWHM linewidth . Calibrated to a peak height of unity, we obtain the Lorentzian lines: When repeating the above Fourier transformation for all the modes with mode index in the resonator, one obtains the full mode spectrum of the resonator. Since the linewidth and the free spectral range are independent